<!DOCTYPE html><html lang="zh-CN" data-theme="light"><head><meta charset="UTF-8"><meta http-equiv="X-UA-Compatible" content="IE=edge"><meta name="viewport" content="width=device-width,initial-scale=1"><title>【学术】4-麦克纳姆轮平台运动学建模 | Jimmy's TechBlog</title><meta name="keywords" content="数学建模,麦克纳姆轮,移动机器人学"><meta name="author" content="哲铭"><meta name="copyright" content="哲铭"><meta name="format-detection" content="telephone=no"><meta name="theme-color" content="#ffffff"><meta name="description" content="对麦克纳姆轮">
<!-- hexo-inject:begin --><!-- hexo-inject:end --><meta property="og:type" content="article">
<meta property="og:title" content="【学术】4-麦克纳姆轮平台运动学建模">
<meta property="og:url" content="https://jimmyliang-lzm.github.io/2021/08/31/mecanum-kinematics/index.html">
<meta property="og:site_name" content="Jimmy&#39;s TechBlog">
<meta property="og:description" content="对麦克纳姆轮">
<meta property="og:locale" content="zh_CN">
<meta property="og:image" content="https://jimmyliang-lzm.github.io/source_storage/mecan_ka.gif">
<meta property="article:published_time" content="2021-09-01T03:00:00.000Z">
<meta property="article:modified_time" content="2021-09-01T04:18:37.068Z">
<meta property="article:author" content="哲铭">
<meta property="article:tag" content="数学建模">
<meta property="article:tag" content="麦克纳姆轮">
<meta property="article:tag" content="移动机器人学">
<meta name="twitter:card" content="summary">
<meta name="twitter:image" content="https://jimmyliang-lzm.github.io/source_storage/mecan_ka.gif"><link rel="shortcut icon" href="/img/favicon.png"><link rel="canonical" href="https://jimmyliang-lzm.github.io/2021/08/31/mecanum-kinematics/"><link rel="preconnect" href="//cdn.jsdelivr.net"/><link rel="preconnect" href="//hm.baidu.com"/><link rel="preconnect" href="//fonts.googleapis.com" crossorigin=""/><link rel="preconnect" href="//busuanzi.ibruce.info"/><link rel="stylesheet" href="/css/index.css"><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/@fortawesome/fontawesome-free/css/all.min.css" media="print" onload="this.media='all'"><script>var _hmt = _hmt || [];
(function() {
  var hm = document.createElement("script");
  hm.src = "https://hm.baidu.com/hm.js?fec3351aac342e4baac87e889e540445";
  var s = document.getElementsByTagName("script")[0]; 
  s.parentNode.insertBefore(hm, s);
})();
</script><link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Titillium+Web&amp;display=swap" media="print" onload="this.media='all'"><script>const GLOBAL_CONFIG = { 
  root: '/',
  algolia: undefined,
  localSearch: {"path":"search.xml","languages":{"hits_empty":"找不到您查询的内容：${query}"}},
  translate: {"defaultEncoding":2,"translateDelay":0,"msgToTraditionalChinese":"繁","msgToSimplifiedChinese":"簡"},
  noticeOutdate: undefined,
  highlight: {"plugin":"highlighjs","highlightCopy":true,"highlightLang":true,"highlightHeightLimit":false},
  copy: {
    success: '复制成功',
    error: '复制错误',
    noSupport: '浏览器不支持'
  },
  relativeDate: {
    homepage: false,
    post: false
  },
  runtime: '',
  date_suffix: {
    just: '刚刚',
    min: '分钟前',
    hour: '小时前',
    day: '天前',
    month: '个月前'
  },
  copyright: undefined,
  lightbox: 'fancybox',
  Snackbar: undefined,
  source: {
    jQuery: 'https://cdn.jsdelivr.net/npm/jquery@latest/dist/jquery.min.js',
    justifiedGallery: {
      js: 'https://cdn.jsdelivr.net/npm/justifiedGallery/dist/js/jquery.justifiedGallery.min.js',
      css: 'https://cdn.jsdelivr.net/npm/justifiedGallery/dist/css/justifiedGallery.min.css'
    },
    fancybox: {
      js: 'https://cdn.jsdelivr.net/npm/@fancyapps/fancybox@latest/dist/jquery.fancybox.min.js',
      css: 'https://cdn.jsdelivr.net/npm/@fancyapps/fancybox@latest/dist/jquery.fancybox.min.css'
    }
  },
  isPhotoFigcaption: false,
  islazyload: true,
  isanchor: false
}</script><script id="config-diff">var GLOBAL_CONFIG_SITE = {
  title: '【学术】4-麦克纳姆轮平台运动学建模',
  isPost: true,
  isHome: false,
  isHighlightShrink: false,
  isToc: true,
  postUpdate: '2021-09-01 12:18:37'
}</script><noscript><style type="text/css">
  #nav {
    opacity: 1
  }
  .justified-gallery img {
    opacity: 1
  }

  #recent-posts time,
  #post-meta time {
    display: inline !important
  }
</style></noscript><script>(win=>{
    win.saveToLocal = {
      set: function setWithExpiry(key, value, ttl) {
        if (ttl === 0) return
        const now = new Date()
        const expiryDay = ttl * 86400000
        const item = {
          value: value,
          expiry: now.getTime() + expiryDay,
        }
        localStorage.setItem(key, JSON.stringify(item))
      },

      get: function getWithExpiry(key) {
        const itemStr = localStorage.getItem(key)

        if (!itemStr) {
          return undefined
        }
        const item = JSON.parse(itemStr)
        const now = new Date()

        if (now.getTime() > item.expiry) {
          localStorage.removeItem(key)
          return undefined
        }
        return item.value
      }
    }
  
    win.getScript = url => new Promise((resolve, reject) => {
      const script = document.createElement('script')
      script.src = url
      script.async = true
      script.onerror = reject
      script.onload = script.onreadystatechange = function() {
        const loadState = this.readyState
        if (loadState && loadState !== 'loaded' && loadState !== 'complete') return
        script.onload = script.onreadystatechange = null
        resolve()
      }
      document.head.appendChild(script)
    })
  
      win.activateDarkMode = function () {
        document.documentElement.setAttribute('data-theme', 'dark')
        if (document.querySelector('meta[name="theme-color"]') !== null) {
          document.querySelector('meta[name="theme-color"]').setAttribute('content', '#0d0d0d')
        }
      }
      win.activateLightMode = function () {
        document.documentElement.setAttribute('data-theme', 'light')
        if (document.querySelector('meta[name="theme-color"]') !== null) {
          document.querySelector('meta[name="theme-color"]').setAttribute('content', '#ffffff')
        }
      }
      const t = saveToLocal.get('theme')
    
          if (t === 'dark') activateDarkMode()
          else if (t === 'light') activateLightMode()
        
      const asideStatus = saveToLocal.get('aside-status')
      if (asideStatus !== undefined) {
        if (asideStatus === 'hide') {
          document.documentElement.classList.add('hide-aside')
        } else {
          document.documentElement.classList.remove('hide-aside')
        }
      }
    
    const detectApple = () => {
      if (GLOBAL_CONFIG_SITE.isHome && /iPad|iPhone|iPod|Macintosh/.test(navigator.userAgent)){
        document.documentElement.classList.add('apple')
      }
    }
    detectApple()
    })(window)</script><!-- hexo injector head_end start -->
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css">

<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/hexo-math@4.0.0/dist/style.css">
<!-- hexo injector head_end end --><meta name="generator" content="Hexo 5.3.0"><link rel="alternate" href="/atom.xml" title="Jimmy's TechBlog" type="application/atom+xml"><!-- hexo-inject:begin --><!-- hexo-inject:end -->
</head><body><div id="sidebar"><div id="menu-mask"></div><div id="sidebar-menus"><div class="avatar-img is-center"><img src= "" data-lazy-src="/img/avatar.jpg" onerror="onerror=null;src='/img/friend_404.gif'" alt="avatar"/></div><div class="site-data"><div class="data-item is-center"><div class="data-item-link"><a href="/archives/"><div class="headline">文章</div><div class="length-num">10</div></a></div></div><div class="data-item is-center"><div class="data-item-link"><a href="/tags/"><div class="headline">标签</div><div class="length-num">22</div></a></div></div><div class="data-item is-center"><div class="data-item-link"><a href="/categories/"><div class="headline">分类</div><div class="length-num">5</div></a></div></div></div><hr/><div class="menus_items"><div class="menus_item"><a class="site-page" href="/"><i class="fa-fw fas fa-home"></i><span> 主页君</span></a></div><div class="menus_item"><a class="site-page" href="/archives/"><i class="fa-fw fas fa-archive"></i><span> 时间之线</span></a></div><div class="menus_item"><a class="site-page" href="/tags/"><i class="fa-fw fas fa-tags"></i><span> 标签</span></a></div><div class="menus_item"><a class="site-page" href="/categories/"><i class="fa-fw fas fa-folder-open"></i><span> 分类君</span></a></div><div class="menus_item"><a class="site-page" href="javascript:void(0);"><i class="fa-fw fas fa-list"></i><span> 今日最爱</span><i class="fas fa-chevron-down expand"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/%E9%9F%B3%E4%B9%90"><i class="fa-fw /music/"></i><span> 0</span></a></li><li><a class="site-page child" href="/%E7%85%A7%E7%89%87"><i class="fa-fw /gallery/"></i><span> 1</span></a></li></ul></div><div class="menus_item"><a class="site-page" href="/link/"><i class="fa-fw fas fa-link"></i><span> 友链</span></a></div><div class="menus_item"><a class="site-page" href="/about/"><i class="fa-fw fas fa-heart"></i><span> 我♂</span></a></div></div></div></div><div class="post" id="body-wrap"><header class="post-bg" id="page-header" style="background-image: url('/source_storage/mecan_ka.gif')"><nav id="nav"><span id="blog_name"><a id="site-name" href="/">Jimmy's TechBlog</a></span><div id="menus"><div id="search-button"><a class="site-page social-icon search"><i class="fas fa-search fa-fw"></i><span> 搜索</span></a></div><div class="menus_items"><div class="menus_item"><a class="site-page" href="/"><i class="fa-fw fas fa-home"></i><span> 主页君</span></a></div><div class="menus_item"><a class="site-page" href="/archives/"><i class="fa-fw fas fa-archive"></i><span> 时间之线</span></a></div><div class="menus_item"><a class="site-page" href="/tags/"><i class="fa-fw fas fa-tags"></i><span> 标签</span></a></div><div class="menus_item"><a class="site-page" href="/categories/"><i class="fa-fw fas fa-folder-open"></i><span> 分类君</span></a></div><div class="menus_item"><a class="site-page" href="javascript:void(0);"><i class="fa-fw fas fa-list"></i><span> 今日最爱</span><i class="fas fa-chevron-down expand"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/%E9%9F%B3%E4%B9%90"><i class="fa-fw /music/"></i><span> 0</span></a></li><li><a class="site-page child" href="/%E7%85%A7%E7%89%87"><i class="fa-fw /gallery/"></i><span> 1</span></a></li></ul></div><div class="menus_item"><a class="site-page" href="/link/"><i class="fa-fw fas fa-link"></i><span> 友链</span></a></div><div class="menus_item"><a class="site-page" href="/about/"><i class="fa-fw fas fa-heart"></i><span> 我♂</span></a></div></div><div id="toggle-menu"><a class="site-page"><i class="fas fa-bars fa-fw"></i></a></div></div></nav><div id="post-info"><h1 class="post-title">【学术】4-麦克纳姆轮平台运动学建模</h1><div id="post-meta"><div class="meta-firstline"><span class="post-meta-date"><i class="far fa-calendar-alt fa-fw post-meta-icon"></i><span class="post-meta-label">发表于</span><time class="post-meta-date-created" datetime="2021-09-01T03:00:00.000Z" title="发表于 2021-09-01 11:00:00">2021-09-01</time><span class="post-meta-separator">|</span><i class="fas fa-history fa-fw post-meta-icon"></i><span class="post-meta-label">更新于</span><time class="post-meta-date-updated" datetime="2021-09-01T04:18:37.068Z" title="更新于 2021-09-01 12:18:37">2021-09-01</time></span><span class="post-meta-categories"><span class="post-meta-separator">|</span><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/%E5%AD%A6%E4%B9%A0%E6%89%8B%E8%AE%B0/">学习手记</a></span></div><div class="meta-secondline"><span class="post-meta-separator">|</span><span class="post-meta-pv-cv" id="" data-flag-title="【学术】4-麦克纳姆轮平台运动学建模"><i class="far fa-eye fa-fw post-meta-icon"></i><span class="post-meta-label">阅读量:</span><span id="busuanzi_value_page_pv"></span></span></div></div></div></header><main class="layout" id="content-inner"><div id="post"><article class="post-content" id="article-container"><h2 id="1-单个麦克纳姆轮运动模型"><a href="#1-单个麦克纳姆轮运动模型" class="headerlink" title="1. 单个麦克纳姆轮运动模型"></a>1. 单个麦克纳姆轮运动模型</h2><p>麦克纳姆轮由一个轮毂与多个辊子构成的异形轮，其中麦克纳姆轮轮轴与辊子滚轴夹角为45°。如图1.1(a)所示，在麦克纳姆轮转动时，辊子将与地面进行摩擦并产生特殊的速度分量${v_{ir}}$，为垂直于辊子的速度<sup>[22]</sup>。</p>
<!-- hexo-inject:begin --><!-- hexo-inject:end --><p><img src= "" data-lazy-src="/2021/08/31/mecanum-kinematics/model.png" alt="图1.1 单个麦克纳姆轮模型参数"></p>
<p>可以根据图1.1(b)可以计算出车轮$i$产生的前向速度分量${w_{Ei}}$与自由辊子与地面所接触的切向速度${v_{ir}}$，并满足公式(4-1)。</p>
$$
{v_{ir}} = \frac{1}{{\cos {{45}^ \circ }}}{r_r}{\omega _i},\:{w_{Ei}} = {r_i}{\omega _i},\:i = 1,2,3,4\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(4-1)
$$
<p>同时根据图1.1(b)可以求出单个轮子的平面坐标系${S_i}{P_i}{E_i}$上的速度分量，他们分别用${w_{Si}}$与${w_{Ei}}$来表示，并整理出式(4-2)。</p>
$$
\left[ {\begin{array}{*{20}{c}}
{{v_{Si}}}\\
{{v_{Ei}}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
0&{\sin {\gamma _i}}\\
{{r_i}}&{\cos {\gamma _i}}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{\omega _i}}\\
{{v_{ir}}}
\end{array}} \right]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(4-2)
$$
<h2 id="2-单个麦轮运动模型转移"><a href="#2-单个麦轮运动模型转移" class="headerlink" title="2. 单个麦轮运动模型转移"></a>2. 单个麦轮运动模型转移</h2><p>在上一节中我们得到了单个麦克纳姆轮的运动模型和速度分量，现在我们将它的速度分量转移到平台中心坐标系。根据图1.1(a)中可以将单个轮子的平面坐标系转变为移动机器人中心点的坐标系${X_R}O{Y_R}$，并得到机器人平台速度分量${v_{Xi}}$与${v_{Yi}}$的关系式(4-3)。</p>
$$
\left[ {\begin{array}{*{20}{c}}
{{v_{Xi}}}\\
{{v_{Yi}}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{\cos {\beta _i}}&{ - \sin {\beta _i}}\\
{\sin {\beta _i}}&{\cos {\beta _i}}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{v_{Si}}}\\
{{v_{Ei}}}
\end{array}} \right]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(4-3)
$$
<p>在本移动机器人平台中，机器人的运动是处于平面上的，根据机器人运动学可以将式(4-3)化简为(4-4)，其中${v_X}$和${v_Y}$表示机器人${X_R}O{Y_R}$坐标系平面内移动的速度分量，${\omega _R}$表示围绕机器人中心点平面内旋转的角速度。</p>
$$
\left[ {\begin{array}{*{20}{c}}
{{v_{Xi}}}\\
{{v_{Yi}}}
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{\begin{array}{*{20}{c}}
1&0&{ - {l_{iy}}}
\end{array}}\\
{\begin{array}{*{20}{c}}
0&1&{{l_{ix}}}
\end{array}}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{v_X}}\\
{{v_Y}}\\
{{\omega _R}}
\end{array}} \right]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(4-4)
$$

<p>通过关系式(4-2)、(4-3)与(4-4)可以得到移动机器人的变换矩阵。</p>
$$
{}^{{w_i}}{T_{{P_i}}} = \left[ {\begin{array}{*{20}{c}}
0&{\sin {\gamma _i}}\\
{{r_i}}&{\cos {\gamma _i}}
\end{array}} \right]
,\:
{}^{{P_i}}{T_R} = \left[ {\begin{array}{*{20}{c}}
{\cos {\beta _i}}&{ - \sin {\beta _i}}\\
{\sin {\beta _i}}&{\cos {\beta _i}}
\end{array}} \right]
,\:
T' = \left[ {\begin{array}{*{20}{c}}
{\begin{array}{*{20}{c}}
1&0&{ - {l_{iy}}}
\end{array}}\\
{\begin{array}{*{20}{c}}
0&1&{{l_{ix}}}
\end{array}}
\end{array}} \right]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(4-5)
$$

$$
{r_i} \ne 0,\:0 < |{\gamma _i}| < \frac{\pi }{2},\:\det ({}^{{P_i}}{T_R}) \ne 0,\:\det ({}^{{w_i}}{T_{{P_i}}}) \ne 0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(4-6)
$$

<p>当公式(4-6)成立时，可以得到矩阵变换关系式(4-7)。</p>
$$
\left[ {\begin{array}{*{20}{c}}
{{\omega _i}}\\
{{v_{ir}}}
\end{array}} \right] = {}^{{w_i}}{T_{{P_i}}}^{ - 1} \cdot {}^{{P_i}}{T_R}^{ - 1} \cdot T'\left[ {\begin{array}{*{20}{c}}
{{v_X}}\\
{{v_Y}}\\
{{\omega _R}}
\end{array}} \right],\:i = 1,2,3,4.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(4-7)
$$
<p>在移动机器人平台的设计中，四个轮子的半径${r_i}$是一致的，可以用$r$来表示四个轮子的半径，经过整理可得关系式(4-8)。</p>
$$
\left[ {\begin{array}{*{20}{c}}
{{\omega _1}}\\
{{\omega _2}}\\
{{\omega _3}}\\
{{\omega _4}}
\end{array}} \right] = \frac{{ - 1}}{r}\left[ {\begin{array}{*{20}{c}}
{\begin{array}{*{20}{c}}
{\frac{{\cos ({\beta _1} - {\gamma _1})}}{{\sin {\gamma _1}}}}\\
{\frac{{\cos ({\beta _2} - {\gamma _2})}}{{\sin {\gamma _2}}}}\\
{\frac{{\cos ({\beta _3} - {\gamma _3})}}{{\sin {\gamma _3}}}}\\
{\frac{{\cos ({\beta _4} - {\gamma _4})}}{{\sin {\gamma _4}}}}
\end{array}}&{\begin{array}{*{20}{c}}
{\frac{{sin({\beta _1} - {\gamma _1})}}{{\sin {\gamma _1}}}}\\
{\frac{{sin({\beta _2} - {\gamma _2})}}{{\sin {\gamma _2}}}}\\
{\frac{{sin({\beta _3} - {\gamma _3})}}{{\sin {\gamma _3}}}}\\
{\frac{{sin({\beta _4} - {\gamma _4})}}{{\sin {\gamma _4}}}}
\end{array}}&{\begin{array}{*{20}{c}}
{\frac{{{l_1}sin({\beta _1} - {\gamma _1} - {\alpha _1})}}{{\sin {\gamma _1}}}}\\
{\frac{{{l_2}sin({\beta _2} - {\gamma _2} - {\alpha _2})}}{{\sin {\gamma _2}}}}\\
{\frac{{{l_3}sin({\beta _3} - {\gamma _3} - {\alpha _3})}}{{\sin {\gamma _3}}}}\\
{\frac{{{l_4}sin({\beta _4} - {\gamma _4} - {\alpha _4})}}{{\sin {\gamma _4}}}}
\end{array}}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{v_X}}\\
{{v_Y}}\\
{{\omega _R}}
\end{array}} \right]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(4-8)
$$

<h2 id="3-通过装配确定最终运动模型"><a href="#3-通过装配确定最终运动模型" class="headerlink" title="3. 通过装配确定最终运动模型"></a>3. 通过装配确定最终运动模型</h2><p>在此给出已经装配好的4-麦克纳姆轮平台的示例，如图3.1所示。</p>
<p><img src= "" data-lazy-src="/2021/08/31/mecanum-kinematics/setup.png" alt="3.1 平台安装示例"></p>
<p>参照图3.1所示移动机器人平台的安装参数，并带入到公式(4-8)中，经过整理最后得到公式(4-9)。</p>
$$
\left[ {\begin{array}{*{20}{c}}
{{\omega _1}}\\
{{\omega _2}}\\
{{\omega _3}}\\
{{\omega _4}}
\end{array}} \right] = \frac{1}{r}\left[ {\begin{array}{*{20}{c}}
{\begin{array}{*{20}{c}}
1\\
1\\
1\\
1
\end{array}}&{\begin{array}{*{20}{c}}
{ - 1}\\
1\\
1\\
{ - 1}
\end{array}}&{\begin{array}{*{20}{c}}
{ - ({l_x} + {l_y})}\\
{({l_x} + {l_y})}\\
{ - ({l_x} + {l_y})}\\
{({l_x} + {l_y})}
\end{array}}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{v_X}}\\
{{v_Y}}\\
{{\omega _R}}
\end{array}} \right]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(4-9)
$$
<p>在此设定一个速度矢量$\vec V$，如图3.2所示。其中${v_X}$表示移动平台在<em>X</em>轴上的速度分量，${v_Y}$为移动平台在<em>Y</em>轴上的速度分量。</p>
<p><img src= "" data-lazy-src="/2021/08/31/mecanum-kinematics/mcm.svg" alt="图3.2 移动速度矢量"></p>
<p>将速度分量${v_X}$与${v_Y}$分别代入式(4-10)可得到四个麦克纳姆轮的转速值，因不包含偏航角速度，${\omega _R}$为零。经过整理可以得到四个麦克纳姆轮的转速，分别用${\omega _1}$~${\omega _4}$表示，得到式(4-11)。</p>
$$
\left\{ {\begin{array}{*{20}{c}}
{{v_1} = r{\omega _1} = {v_X} - {v_Y} - ({l_x} + {l_y}){\omega _R}}\\
{{v_2} = r{\omega _2} = {v_X} + {v_Y} + ({l_x} + {l_y}){\omega _R}}\\
{{v_3} = r{\omega _3} = {v_X} + {v_Y} - ({l_x} + {l_y}){\omega _R}}\\
{{v_4} = r{\omega _4} = {v_X} - {v_Y} + ({l_x} + {l_y}){\omega _R}}
\end{array}} \right.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(4-10)
$$

$$
\left\{ {\begin{array}{*{20}{c}}
{{\omega _1} = ({L_X} - {L_Y})/r}\\
{{\omega _2} = ({L_X} + {L_Y})/r}\\
{{\omega _3} = ({L_X} + {L_Y})/r}\\
{{\omega _4} = ({L_X} - {L_Y})/r}
\end{array}} \right.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:(4-11)
$$

<p>通过公式(4-10)对移动机器人平台进行验证即可得到如图3.3所示的平台运动状态，并最终确定建立的机器人移动平台数学模型正确。</p>
<p><img src= "" data-lazy-src="/2021/08/31/mecanum-kinematics/motation.svg" alt="图3.3 平台全向运动示意"></p>
<h2 id="参考文献"><a href="#参考文献" class="headerlink" title="参考文献"></a>参考文献</h2><ol>
<li>Taheri H, Qiao B, Ghaeminezhad N. Kinematic model of a four mecanum wheeled mobile robot[J]. International journal of computer applications, 2015, 113(3): 6-9.</li>
<li>孙全胜. 基于STM32单片机的麦克纳姆轮小车设计[J]. 现代信息科技, 2019, 003(022):P.174-175.</li>
</ol>
</article><div class="post-copyright"><div class="post-copyright__author"><span class="post-copyright-meta">文章作者: </span><span class="post-copyright-info"><a href="mailto:undefined">哲铭</a></span></div><div class="post-copyright__type"><span class="post-copyright-meta">文章链接: </span><span class="post-copyright-info"><a href="https://jimmyliang-lzm.github.io/2021/08/31/mecanum-kinematics/">https://jimmyliang-lzm.github.io/2021/08/31/mecanum-kinematics/</a></span></div><div class="post-copyright__notice"><span class="post-copyright-meta">版权声明: </span><span class="post-copyright-info">本博客所有文章除特别声明外，均采用 <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" target="_blank">CC BY-NC-SA 4.0</a> 许可协议。转载请注明来自 <a href="https://jimmyliang-lzm.github.io" target="_blank">Jimmy's TechBlog</a>！</span></div></div><div class="tag_share"><div class="post-meta__tag-list"><a class="post-meta__tags" href="/tags/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/">数学建模</a><a class="post-meta__tags" href="/tags/%E9%BA%A6%E5%85%8B%E7%BA%B3%E5%A7%86%E8%BD%AE/">麦克纳姆轮</a><a class="post-meta__tags" href="/tags/%E7%A7%BB%E5%8A%A8%E6%9C%BA%E5%99%A8%E4%BA%BA%E5%AD%A6/">移动机器人学</a></div><div class="post_share"><div class="social-share" data-image="/source_storage/mecan_ka.gif" data-sites="facebook,twitter,wechat,weibo,qq"></div><link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/social-share.js/dist/css/share.min.css" media="print" onload="this.media='all'"><script src="https://cdn.jsdelivr.net/npm/social-share.js/dist/js/social-share.min.js" defer></script></div></div><nav class="pagination-post" id="pagination"><div class="prev-post pull-left"><a href="/2021/10/05/Get_bilibili_cookie/"><img class="prev-cover" src= "" data-lazy-src="/source_storage/2233.jpg" onerror="onerror=null;src='/img/404.jpg'" alt="cover of previous post"><div class="pagination-info"><div class="label">上一篇</div><div class="prev_info">【教程】如何获取B站用户Cookie</div></div></a></div><div class="next-post pull-right"><a href="/2021/02/09/Java-Note-01/"><img class="next-cover" src= "" data-lazy-src="/source_storage/java.jpg" onerror="onerror=null;src='/img/404.jpg'" alt="cover of next post"><div class="pagination-info"><div class="label">下一篇</div><div class="next_info">【Java学习】Java入门笔记1</div></div></a></div></nav></div><div class="aside-content" id="aside-content"><div class="card-widget card-info"><div class="is-center"><div class="avatar-img"><img src= "" data-lazy-src="/img/avatar.jpg" onerror="this.onerror=null;this.src='/img/friend_404.gif'" alt="avatar"/></div><div class="author-info__name">哲铭</div><div class="author-info__description">哲铭的个人站点，分享技术学习心得，项目开发心路历程，以及日常沙雕操作，欢迎访问！</div></div><div class="card-info-data"><div class="card-info-data-item is-center"><a href="/archives/"><div class="headline">文章</div><div class="length-num">10</div></a></div><div class="card-info-data-item is-center"><a href="/tags/"><div class="headline">标签</div><div class="length-num">22</div></a></div><div class="card-info-data-item is-center"><a href="/categories/"><div class="headline">分类</div><div class="length-num">5</div></a></div></div><a class="button--animated" id="card-info-btn" target="_blank" rel="noopener" href="https://github.com/xxxxxx"><i class="fab fa-github"></i><span>Follow Me</span></a><div class="card-info-social-icons is-center"><a class="social-icon" href="https://github.com/JimmyLiang-lzm" target="_blank" title="Github"><i class="fab fa-github"></i></a><a class="social-icon" href="mailto:mibemail@163.com" target="_blank" title="Email"><i class="fas fa-envelope"></i></a></div></div><div class="card-widget card-announcement"><div class="item-headline"><i class="fas fa-bullhorn card-announcement-animation"></i><span>公告</span></div><div class="announcement_content">本博客网站因资金原因暂时托管到Github与Gitee上，当发工资的时候会搭建自己的服务器哒🤪！<a target="_blank" rel="noopener" href="https://zmtechn.gitee.io"><button>Gitee访问地址</button></a><a href="https://jimmyliang-lzm.github.io"><button>Github访问地址</button></a></div></div><div class="sticky_layout"><div class="card-widget" id="card-toc"><div class="item-headline"><i class="fas fa-stream"></i><span>目录</span></div><div class="toc-content"><ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#1-%E5%8D%95%E4%B8%AA%E9%BA%A6%E5%85%8B%E7%BA%B3%E5%A7%86%E8%BD%AE%E8%BF%90%E5%8A%A8%E6%A8%A1%E5%9E%8B"><span class="toc-text">1. 单个麦克纳姆轮运动模型</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#2-%E5%8D%95%E4%B8%AA%E9%BA%A6%E8%BD%AE%E8%BF%90%E5%8A%A8%E6%A8%A1%E5%9E%8B%E8%BD%AC%E7%A7%BB"><span class="toc-text">2. 单个麦轮运动模型转移</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#3-%E9%80%9A%E8%BF%87%E8%A3%85%E9%85%8D%E7%A1%AE%E5%AE%9A%E6%9C%80%E7%BB%88%E8%BF%90%E5%8A%A8%E6%A8%A1%E5%9E%8B"><span class="toc-text">3. 通过装配确定最终运动模型</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE"><span class="toc-text">参考文献</span></a></li></ol></div></div><div class="card-widget card-recent-post"><div class="item-headline"><i class="fas fa-history"></i><span>最新文章</span></div><div class="aside-list"><div class="aside-list-item"><a class="thumbnail" href="/2021/10/05/Get_bilibili_cookie/" title="【教程】如何获取B站用户Cookie"><img src= "" data-lazy-src="/source_storage/2233.jpg" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="【教程】如何获取B站用户Cookie"/></a><div class="content"><a class="title" href="/2021/10/05/Get_bilibili_cookie/" title="【教程】如何获取B站用户Cookie">【教程】如何获取B站用户Cookie</a><time datetime="2021-10-06T03:00:00.000Z" title="发表于 2021-10-06 11:00:00">2021-10-06</time></div></div><div class="aside-list-item"><a class="thumbnail" href="/2021/08/31/mecanum-kinematics/" title="【学术】4-麦克纳姆轮平台运动学建模"><img src= "" data-lazy-src="/source_storage/mecan_ka.gif" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="【学术】4-麦克纳姆轮平台运动学建模"/></a><div class="content"><a class="title" href="/2021/08/31/mecanum-kinematics/" title="【学术】4-麦克纳姆轮平台运动学建模">【学术】4-麦克纳姆轮平台运动学建模</a><time datetime="2021-09-01T03:00:00.000Z" title="发表于 2021-09-01 11:00:00">2021-09-01</time></div></div><div class="aside-list-item"><a class="thumbnail" href="/2021/02/09/Java-Note-01/" title="【Java学习】Java入门笔记1"><img src= "" data-lazy-src="/source_storage/java.jpg" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="【Java学习】Java入门笔记1"/></a><div class="content"><a class="title" href="/2021/02/09/Java-Note-01/" title="【Java学习】Java入门笔记1">【Java学习】Java入门笔记1</a><time datetime="2021-02-10T04:06:00.000Z" title="发表于 2021-02-10 12:06:00">2021-02-10</time></div></div><div class="aside-list-item"><a class="thumbnail" href="/2021/01/31/PID-DCMotor-Simulate/" title="【学习手记】基于AT89C52单片机的PID直流电机控制系统仿真"><img src= "" data-lazy-src="/source_storage/Proteus.jpg" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="【学习手记】基于AT89C52单片机的PID直流电机控制系统仿真"/></a><div class="content"><a class="title" href="/2021/01/31/PID-DCMotor-Simulate/" title="【学习手记】基于AT89C52单片机的PID直流电机控制系统仿真">【学习手记】基于AT89C52单片机的PID直流电机控制系统仿真</a><time datetime="2021-01-31T12:50:00.000Z" title="发表于 2021-01-31 20:50:00">2021-01-31</time></div></div><div class="aside-list-item"><a class="thumbnail" href="/2021/01/29/Matlab_Robot_simulation/" title="【学习手记】MATLAB与机器人仿真项目训练"><img src= "" data-lazy-src="/source_storage/matlabrobotsimulate.png" onerror="this.onerror=null;this.src='/img/404.jpg'" alt="【学习手记】MATLAB与机器人仿真项目训练"/></a><div class="content"><a class="title" href="/2021/01/29/Matlab_Robot_simulation/" title="【学习手记】MATLAB与机器人仿真项目训练">【学习手记】MATLAB与机器人仿真项目训练</a><time datetime="2021-01-30T05:14:00.000Z" title="发表于 2021-01-30 13:14:00">2021-01-30</time></div></div></div></div></div></div></main><footer id="footer"><div id="footer-wrap"><div class="copyright">&copy;2020 - 2021 By 哲铭</div><div class="framework-info"><span>框架 </span><a target="_blank" rel="noopener" href="https://hexo.io">Hexo</a><span class="footer-separator">|</span><span>主题 </span><a target="_blank" rel="noopener" href="https://github.com/jerryc127/hexo-theme-butterfly">Butterfly</a></div></div></footer></div><div id="rightside"><div id="rightside-config-hide"><button id="readmode" type="button" title="阅读模式"><i class="fas fa-book-open"></i></button><button id="translateLink" type="button" title="简繁转换">繁</button><button id="darkmode" type="button" title="浅色和深色模式转换"><i class="fas fa-adjust"></i></button><button id="hide-aside-btn" type="button" title="单栏和双栏切换"><i class="fas fa-arrows-alt-h"></i></button></div><div id="rightside-config-show"><button id="rightside_config" type="button" title="设置"><i class="fas fa-cog fa-spin"></i></button><button class="close" id="mobile-toc-button" type="button" title="目录"><i class="fas fa-list-ul"></i></button><button id="go-up" type="button" title="回到顶部"><i class="fas fa-arrow-up"></i></button></div></div><div id="local-search"><div class="search-dialog"><div class="search-dialog__title" id="local-search-title">本地搜索</div><div id="local-input-panel"><div id="local-search-input"><div class="local-search-box"><input class="local-search-box--input" placeholder="搜索文章" type="text"/></div></div></div><hr/><div id="local-search-results"></div><span class="search-close-button"><i class="fas fa-times"></i></span></div><div id="search-mask"></div></div><div><script src="/js/utils.js"></script><script src="/js/main.js"></script><script src="/js/tw_cn.js"></script><script src="https://cdn.jsdelivr.net/npm/instant.page/instantpage.min.js" type="module"></script><script src="https://cdn.jsdelivr.net/npm/vanilla-lazyload/dist/lazyload.iife.min.js"></script><script src="/js/search/local-search.js"></script><div class="js-pjax"><script>if (!window.MathJax) {
  window.MathJax = {
    tex: {
      inlineMath: [ ['$','$'], ["\\(","\\)"]],
      tags: 'ams'
    },
    chtml: {
      scale: 1.2
    },
    options: {
      renderActions: {
        findScript: [10, doc => {
          for (const node of document.querySelectorAll('script[type^="math/tex"]')) {
            const display = !!node.type.match(/; *mode=display/)
            const math = new doc.options.MathItem(node.textContent, doc.inputJax[0], display)
            const text = document.createTextNode('')
            node.parentNode.replaceChild(text, node)
            math.start = {node: text, delim: '', n: 0}
            math.end = {node: text, delim: '', n: 0}
            doc.math.push(math)
          }
        }, ''],
        insertScript: [200, () => {
          document.querySelectorAll('mjx-container:not\([display]\)').forEach(node => {
            const target = node.parentNode
            if (target.nodeName.toLowerCase() === 'li') {
              target.parentNode.classList.add('has-jax')
            } else {
              target.classList.add('has-jax')
            }
          });
        }, '', false]
      }
    }
  }
  
  const script = document.createElement('script')
  script.src = 'https://cdn.jsdelivr.net/npm/mathjax/MathJax.js?config=TeX-AMS-MML_HTMLorMML'
  script.id = 'MathJax-script'
  script.async = true
  document.head.appendChild(script)
} else {
  MathJax.startup.document.state(0)
  MathJax.texReset()
  MathJax.typeset()
}</script></div><script defer="defer" id="ribbon" src="/js/third-party/canvas-ribbon.js" size="150" alpha="0.6" zIndex="-1" mobile="true" data-click="true"></script><script async data-pjax src="//busuanzi.ibruce.info/busuanzi/2.3/busuanzi.pure.mini.js"></script></div><!-- hexo-inject:begin --><!-- hexo-inject:end --></body></html>